Finite Group Databasebeta

← #27413658

Subgroups — #27413658

Order 13568.

874 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810730C211yesmaximal, characteristic, complemented
2C2×#16810677C211yesmaximal, characteristic
3C22⋊Q8×Dic53C211yesmaximal, characteristic
4C22⋊Q8×Dic53C211yesmaximal, characteristic
5#20276215C211yesmaximal, characteristic, complemented
6#20276148C211yesmaximal, characteristic
7#20276106C211yesmaximal, characteristic, complemented
8#20276092C211yesmaximal, characteristic, complemented
9#20276035C211yesmaximal, characteristic, complemented
10#20275915C211yesmaximal, characteristic
11#20275908C211yesmaximal, characteristic
12#20274611C211yesmaximal, characteristic, complemented
13#20274547C211yesmaximal, characteristic, complemented
14#20274547C211yesmaximal, characteristic, complemented
15C53×#2001C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C22×C4×Dic53C2211yescharacteristic
19#16810730—24no—
20#16810677—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×C22⋊#88322C2211yescharacteristic
31C2×#13133127C2211yescharacteristic
32C2×#13133127C2211yescharacteristic
33#16810530C2211yescharacteristic
34#16810530C2211yescharacteristic
35#16810522C2211yescharacteristic
36#16810522C2211yescharacteristic
37#16810509—21no—
38#16810509—21no—
39#16810498—21no—
40#16810498—21no—
41#16810483C2211yescharacteristic
42#16810483C2211yescharacteristic
43C4⋊C4×Dic53—21no—
44C4⋊C4×Dic53—21no—
45C4⋊C4×Dic53C2211yescharacteristic
46C4⋊C4×Dic53C2211yescharacteristic
47C22⋊C4×Dic53—21no—
48C22⋊C4×Dic53—21no—
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54C22×Q16×C53C2211yescharacteristic
55C2×C22⋊Q8×C53C2211yescharacteristic
56C2×C22⋊Q8×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×#103×C53C2211yescharacteristic
59C2×C22⋊C8×C53C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C23×Dic53D411yescharacteristic
62C23×Dic53D411yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65Q8×Dic53—22no—
66Q8×Dic53—22no—
67Q8×Dic53—22no—
68Q8×Dic53—22no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C4×Dic53—22no—
80C2×C4×Dic53—22no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89#13133150—22no—
90#13133149—22no—
91#13133148—22no—
92#13133146—22no—
93#13133146—22no—
94#13133141—22no—
95#13133141—22no—
96#13133141—22no—
97#13133141—22no—
98C22⋊#88322—22no—
99#13133127—22no—
100#13133127D414yes—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111C22×Q8×C53C2311yescharacteristic
112C22×Q8×C53C2311yescharacteristic
113C23×C4×C53C2311yescharacteristic
114C2×Q16×C53—22no—
115C2×Q16×C53—22no—
116C2×Q16×C53—22no—
117C22×C8×C53C2311yescharacteristic
118C22×C8×C53C2311yescharacteristic
119C22⋊Q8×C53—22no—
120C22⋊Q8×C53—22no—
121C22⋊Q8×C53—22no—
122C22⋊Q8×C53—22no—
123C2×C4⋊C4×C53—21no—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C22⋊C4×C53—21no—
128C2×C22⋊C4×C53—21no—
129#103×C53—22no—
130#103×C53—22no—
131C22⋊C8×C53—22no—
132C24×C53C2×D411yescharacteristic
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53#4112yes—
144C2×Q8×C53#4112yes—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2411yescharacteristic
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—41no—
166C22×Dic53—41no—
167C22×Dic53C2×D411yescharacteristic
168C22×Dic53C2×D411yescharacteristic
169C22×Dic53C2×D411yescharacteristic
170C22×Dic53C2×D411yescharacteristic
171Q16×C53—28no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53C2×D412yes—
178C4⋊C4×C53—24no—
179C4⋊C4×C53—24no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53#4112yes—
183C4⋊C4×C53#4112yes—
184C22⋊C4×C53—24no—
185C22⋊C4×C53—24no—
186C22⋊Dic53—42no—
187C22⋊Dic53—42no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—42no—
197C4⋊Dic53—42no—
198C4⋊Dic53—42no—
199C4⋊Dic53—42no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—22no—
203C4×Dic53—22no—
204C4×Dic53—42no—
205C4×Dic53—42no—
206C4×Dic53—42no—
207C4×Dic53—42no—
208C4×Dic53C2×D412yes—
209C4×Dic53#4112yes—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322C2×D412yes—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53#13712yes—
223C23×C53C22×D411yescharacteristic
224Q8×C53—22no—
225Q8×C53—22no—
226Q8×C53—22no—
227Q8×C53—22no—
228Q8×C53—41no—
229Q8×C53—41no—
230Q8×C53—41no—
231Q8×C53—41no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53C22×D411yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—44no—
267C2×Dic53—44no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53#12112yes—
281C2×Dic53#12112yes—
282C2×Dic53#13712yes—
283C8×C53—22no—
284C8×C53—41no—
285#88322—22no—
286#88322—41no—
287#23709—531nomaximal, Sylow
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53C2×#12111yescharacteristic
296C22×C53#52511yescharacteristic
297C22×C53#52511yescharacteristic
298C22×C53C2×#13711yescharacteristic
299C22×C53C2×#13711yescharacteristic
300C22×C53#57511yescharacteristic
301C22×C53#57511yescharacteristic
302C4×C53—42no—
303C4×C53—42no—
304C4×C53—41no—
305C4×C53—42no—
306C4×C53—42no—
307C4×C53#52511yescharacteristic
308C4×C53#52911yescharacteristic
309C4×C53#52911yescharacteristic
310C4×C53#53211yescharacteristic
311Dic53—22no—
312Dic53—22no—
313Dic53—22no—
314Dic53—41no—
315Dic53—81no—
316Dic53—81no—
317Dic53—81no—
318Dic53—81no—
319C2×#441—531no—
320C2×#432—531no—
321C4×C22⋊Q8—531no—
322C4×C22⋊Q8—531no—
323#2088—531no—
324#2077—531no—
325#2068—531no—
326#2066—531no—
327#2056—531no—
328#2036—531no—
329#2035—531no—
330#2001D5311yescharacteristic
331#1776—531no—
332#1763—531no—
333#1763—531no—
334C2×C53—42no—
335C2×C53#239211yescharacteristic
336C2×C53#305611yescharacteristic
337C2×C53#305611yescharacteristic
338C2×C53#311111yescharacteristic
339C2×C53#311311yescharacteristic
340C2×C53#311611yescharacteristic
341C2×C53#311711yescharacteristic
342C22×Q16C2×D5311yescharacteristic
343C2×C22⋊Q8C2×D5311yescharacteristic
344C2×C22⋊Q8C2×D5311yescharacteristic
345C2×C4×Q8—531no—
346C2×C4×Q8—531no—
347C22×C42—531no—
348#441—1064no—
349#432—1064no—
350C2×#107—531no—
351C2×#106—531no—
352C2×#106—531no—
353C2×C4⋊C8—531no—
354C2×C4⋊C8—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×#103C2×D5311yescharacteristic
360C2×#103C2×D5311yescharacteristic
361C2×C22⋊C8—531no—
362C2×C22⋊C8C2×D5311yescharacteristic
363C2×#97—531no—
364C2×#97—531no—
365#391—531no—
366#391—531no—
367#389—531no—
368#389—531no—
369#387—1061no—
370#387—1061no—
371#385—1061no—
372#385—1061no—
373#380—531no—
374#380—531no—
375C4×C4⋊C4—531no—
376C4×C4⋊C4—531no—
377C4×C4⋊C4—1061no—
378C4×C4⋊C4—1061no—
379C4×C22⋊C4—1061no—
380C4×C22⋊C4—1061no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340C2×D5311yescharacteristic
387C53#2370911yescharacteristic, complemented, Sylow
388C22×Q8C22×D5311yescharacteristic
389C22×Q8C22×D5311yescharacteristic
390C23×C4—531no—
391C23×C4—531no—
392C23×C4C22×D5311yescharacteristic
393C2×Q16—22no—
394C2×Q16—22no—
395C2×Q16—22no—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8C22×D5311yescharacteristic
399C22×C8C22×D5311yescharacteristic
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403C22⋊Q8—22no—
404C4×Q8—1062no—
405C4×Q8—1062no—
406C4×Q8—1062no—
407C4×Q8—1062no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—1062no—
429C2×C42—1062no—
430C2×C42—1061no—
431C2×C42—1061no—
432C2×C42—1061no—
433C2×C42—1061no—
434#107—1062no—
435#106—1062no—
436#106—1062no—
437C4⋊C8—1062no—
438C4⋊C8—1062no—
439#103—22no—
440#103—22no—
441#103—1062no—
442#103—1062no—
443#103—1062no—
444#103—1062no—
445C22⋊C8—22no—
446C22⋊C8—1062no—
447#97—534no—
448#97—1062no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459C24D4×D5311yescharacteristic
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8#1171994712yes—
471C2×Q8#1171994712yes—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—531no—
485C22×C4—531no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4—2121no—
495C22×C4—2121no—
496C22×C4D4×D5311yescharacteristic
497C22×C4C23×D5311yescharacteristic
498Q16—28no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—532no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8D4×D5312yes—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4#1171994712yes—
528C4⋊C4#1171994712yes—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—2122no—
532C22⋊C4—2122no—
533C42—532no—
534C42—532no—
535C42—1062no—
536C42—1062no—
537C42—1061no—
538C42—1061no—
539C42—2122no—
540C42—2122no—
541C42—2122no—
542C42—2122no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23#1313324912yes—
550C23C2×D4×D5311yescharacteristic
551Q8—22no—
552Q8—22no—
553Q8—22no—
554Q8—22no—
555Q8—41no—
556Q8—41no—
557Q8—41no—
558Q8—41no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—532no—
579C2×C4—532no—
580C2×C4—532no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—2124no—
591C2×C4—2124no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4C2×D4×D5311yescharacteristic
605C2×C4#41×D5311yescharacteristic
606C2×C4C2×#1171994711yescharacteristic
607C2×C4C2×#1171994711yescharacteristic
608C2×C4C2×#1171994711yescharacteristic
609C2×C4C2×#1171994711yescharacteristic
610C8—22no—
611C8—41no—
612C8—1062no—
613C8—2121no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22#121×D5311yescharacteristic
622C22#1681112911yescharacteristic
623C22#1681112911yescharacteristic
624C22#137×D5311yescharacteristic
625C22C2×#1313324911yescharacteristic
626C22#1681142011yescharacteristic
627C22#1681142211yescharacteristic
628C4—42no—
629C4—41no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—1062no—
634C4—1062no—
635C4—1062no—
636C4—4241no—
637C4—4241no—
638C4—4241no—
639C4—4241no—
640C4—2121no—
641C4#1681112911yescharacteristic
642C4#1681115411yescharacteristic
643C4#1681115411yescharacteristic
644C4#1681117111yescharacteristic
645C2—42no—
646C2#2027795011yescharacteristic
647C2#2028434211yescharacteristic
648C2#2028434411yescharacteristic
649C2#2028482511yescharacteristic
650C2#2028484011yescharacteristic
651C2#2028487011yescharacteristic
652C2#2028488711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2001C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2001D53
hypocenterC53#23709
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16810730C2semidirect
5. #20276215C2semidirect
7. #20276106C2semidirect
8. #20276092C2semidirect
9. #20276035C2semidirect
12. #20274611C2semidirect
13. #20274547C2semidirect
14. #20274547C2semidirect
387. C53#23709semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168107302
C2×#168106772
C22⋊Q8×Dic532
#202762152
#202761482
#202761062
#202760922
#202760352
#202759152
#202759082
#202746112
#202745472
C53×#20012
#2370953